N2018 P2 Q5

N2018 P2 Q5

Junior College 2
7 marks
Free

The manufacturer of a certain type of fan used for cooling electronic devices claims that the mean time to failure (MTTF) is 65 000 hours. The quality control manager suspects that the MTTF is actually less than 65 000 hours and decides to carry out a hypothesis test on a sample of these fans. (An accelerated testing procedure is used to find the MTTF.)

  1. Explain why the manager should take a sample of at least 30 fans, and state how these fans should be chosen.[2]
  2. State suitable hypotheses for the test, defining any symbols that you use.[2]

The quality control manager takes a suitable sample of 43 fans, and finds that they have an MTTF of 64 230 hours.

  1. Given that the manager concludes that there is no reason to reject the null hypothesis at the 5% level of significance, find the range of possible values of the variance used in calculating the test statistic.[3]

Default solution

(i) A sample of at least \(30\) makes the normal approximation to the sample mean reasonable by the central limit theorem. Choose a random, representative sample of fans independently.

(ii) Let \(\mu\) be the population mean time to failure. \(H_0:\mu=65000\) hours; \(H_1:\mu<65000\) hours.

(iii) Let \(s^2\) be the variance used in the statistic. \(z=\frac{64230-65000}{s/\sqrt{43}}=-\frac{770\sqrt{43}}s\).

No rejection at \(5\%\): \(z\ge-1.64485\Rightarrow s^2\ge\left(\frac{770\sqrt{43}}{1.64485}\right)^2\approx9.42\times10^6\text{ hours}^2\).

Answer:(ii) \(H_0:\mu=65000,\ H_1:\mu<65000\); (iii) \(s^2\ge9.42\times10^6\text{ hours}^2\)

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